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A SHORT NOTE ON THE FRAME SET OF ODD FUNCTIONS

Part of: General quantum mechanics and problems of quantization Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  15 August 2018

MARKUS FAULHUBER Open the ORCID record for MARKUS FAULHUBER [Opens in a new window]
MARKUS FAULHUBER*
Affiliation:
Analysis Group, Department of Mathematical Sciences, NTNU Trondheim, Sentralbygg 2, Gløshaugen, Trondheim, Norway email markus.faulhuber@ntnu.no
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Abstract

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We give a simple argument which shows that Gabor systems consisting of oddfunctions of $d$ variables and symplectic lattices of density $2^{d}$ cannot constitute a Gabor frame. In the one-dimensional,separable case, this follows from a more general result of Lyubarskii andNes [‘Gabor frames with rational density’, Appl. Comput. Harmon. Anal.34(3) (2013), 488–494]. We use a differentapproach exploiting the algebraic relation between the ambiguity functionand the Wigner distribution as well as their relation given by the(symplectic) Fourier transform. Also, we do not need the assumption that thelattice is separable and, hence, new restrictions are added to the fullframe set of odd functions.

Keywords

ambiguity functionFeichtinger’s algebraframe setGabor frameWigner distribution

MSC classification

Primary: 42C15: General harmonic expansions, frames
Secondary: 81S30: Phase-space methods including Wigner distributions, etc.

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 98 , Issue 3 , December 2018 , pp. 481 - 493
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

The author was supported by the Erwin Schrödinger program of the Austrian Science Fund (FWF): J4100-N32.

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